This invention relates to computed tomography imaging and especially to the use of multispectral imaging in limited-angle reconstruction.
In some CT applications the object can be scanned only in a limited angular range. For instance, the industrial x-ray CT the object may be long and rectangular such that there is too much attenuation for x-ray beams at large oblique incidence angles, or the object might be obstructed in some angular range. These situations are illustrated in FIGS. 1 and 2. In electron microscopy biological specimens in the form of thin slices can only be scanned by electrons in a limited angular range because of strong attenuation of the electron beam at large oblique incidence angles.
There are many methods for reconstructing images from their projections. One of these is the Fourier method, which involves a transformation of the data from the projections into what is known as Fourier or frequency space. Points in the Fourier space for which there are no data are estimated by interpolation. The reconstruction is then obtained from the Fourier space by taking the inverse Fourier transform. FIG. 3a illustrates in the real, object space a limited angular range of projection data. It has been shown that from such restricted angular data one can calculate only the frequency components in a limited angular range in the frequency space, as illustrated in FIG. 3b. The region in the frequency space where the frequency components of an object are known are referred to as the "allowed cone" and where they are not known as the "missing cone".
In principle, the missing cone frequency components can be recovered and continued from those in the allowed cone through the knowledge of the location and finite extent of the object. There are several techniques for doing this, and in all of them the object is assumed to have finite boundaries and the density outside the boundary is set to zero. This a priori knowledge is coupled with the partial Fourier components in the allowed cone. For example, the frequency components of the object can be expanded in a Fourier series with the coefficients determined from the known extent of the object (see T. Inouye, IEEE Transactions on Nuclear Science, NS-26 (1979) 2666-2669). Another way to achieve this is through the iteration scheme proposed by the inventor, V. Perez-Mendez, and B. Macdonald, IEEE Transactions on Nuclear Science, NS-26 (1979) 2797-2805. The object is transformed back and forth between the object space and the frequency space, being corrected in each step by the finite object extent and the known frequency components. While it is theoretically possible to determine the object exactly, in practice recovery of the missing cone frequency components cannot be done perfectly because the problem is ill conditioned and because of errors in the input data and imprecision in numerical computation.
The principal object of the invention is to improve on this situation. Multiple energy scanning has been successfully used to enhance images in complete-angle scanning, in x-ray fluoroscopy imaging where a patient is imaged in conventional CT mode at different x-ray energies to separate out the iodine component in the image. Refer to S. J. Riederer and C. A. Mistretta, Medical Physics, 4 (1977) 474-481. The present invention is for limited-angle imaging and is distinguishable in other respects. The use of the upper and lower bounds of an object in limited-angle reconstruction has been proposed by A. Lent and H. Tuy, "An Iterative Method for the Extrapolation of Band-Limited Functions", Technical Report No. MIPG 35 (1979), State University of New York at Buffalo. Positivity, the constraint that physical objects have only non-negative density, is a particular case of the constraint of lower boundness. It was found by the inventor and V. Perez-Mendez, Optical Engineering, 20 (1981) 586-589 and other references, that incorporating the constraint of positivity in addition to the constraint of finite object extent produces only a small improvement in the limited-angle reconstruction of extended objects. This prior work on bounds is discussed in more detail later.